Open Coloring Axioms may be viewed as consistent generalizations of Ramsey's Theorem to $\omega_1$ in which topological restrictions are placed on the colorings. The first of these, denoted $\mathsf{OCA}_{ARS}$, appeared in the 1985 paper by Abraham, Rubin, and Shelah. There the authors showed that $\mathsf{OCA}_{ARS}$ is consistent with $\mathsf{ZFC}$. To ensure that the posets which add the homogeneous sets satisfy the c.c.c., they construct a type of ``diagonalization" object (for a continuous coloring $\chi$) called a Preassignment of Colors, which guides the forcing to add the $\chi$-homogeneous sets.

However, the only known constructions of effective preassignments require the $\mathsf{CH}$. Since a forcing iteration of $\aleph_1$-sized posets all of whose proper initial segments satisfy the $\mathsf{CH}$ results in a model in which $2^{\aleph_0}$ is at most $\aleph_2$, this leads naturally to the question of whether $\mathsf{OCA}_{ARS}$ is consistent, say, with $2^{\aleph_0}=\aleph_3$.

In joint work with Itay Neeman, we answer this question in the affirmative. In light of the $\mathsf{CH}$ obstacle, we only construct names for preassignments with respect to a small class $\mathcal{A}$ of $\mathsf{CH}$-preserving iterations. However, our preassignments are powerful enough to work even over models in which the $\mathsf{CH}$ fails.

Our final forcing is built by combining the members of $\mathcal{A}$ into a new type of forcing, called a Partition Product. A partition product is a type of restricted memory iteration with isomorphism and coherent-overlap conditions on the memories. In particular, each ``memory" is isomorphic to a member of $\mathcal{A}$.

In this talk, we will describe in some detail the definition of a Partition Product. We will then discuss how to construct more general preassignments than those used by Abraham, Rubin, and Shelah, gesturing at the end towards the full construction which we use for our theorem.

Speaker Bio: I am a Visiting Assistant Professor in the department of Mathematics at the University of Pittsburgh, having graduated from UCLA in the Fall of 2019 under the supervision of Itay Neeman. I am interested in questions about what combinatorial principles determine the size of the continuum, as well as in questions about the tension between compactness and incompactness principles in set theory. I reside in Pittsburgh with my wife, Marian (who is a philosopher of physics), with our indefatigable toddler Zoe, and with our two cats.